Question

solve the polynomial equation by finding all roots.

1. ( 2x^3 - 3x^2 + 8x - 12 = 0 )
2. ( x^4 - 5x^3 + 3x^2 + x = 0 )

Explanation:

Step1: Group terms for factoring

Group the first two terms and the last two terms: $(2x^3 - 3x^2)+(8x - 12)=0$

Step2: Factor out GCF from each group

Factor out $x^2$ from the first group and $4$ from the second group: $x^2(2x - 3)+4(2x - 3)=0$

Step3: Factor out common binomial

Factor out $(2x - 3)$: $(2x - 3)(x^2 + 4)=0$

Step4: Set each factor to zero

Set $2x - 3 = 0$ and $x^2 + 4 = 0$

Step5: Solve for x in linear equation

For $2x - 3 = 0$, we get $2x=3$ so $x=\frac{3}{2}$

Step6: Solve for x in quadratic equation

For $x^2 + 4 = 0$, we have $x^2=-4$, so $x=\pm 2i$ (since $i^2 = - 1$)

Answer:

The roots of the polynomial equation $2x^3 - 3x^2 + 8x - 12 = 0$ are $x=\frac{3}{2},x = 2i,x=-2i$